(This really belongs in "Off Topic"; it's not a support question.)
That's a much bigger question (or set of questions) than it may appear to be at first blush.
To answer the subject line question directly: yes, it is possible for a sole proprietor to successfully host (or, rather more usually, manage) his or her own web site, with a few caveats. First, it depends upon the nature of the site. Something that is little more than a "point of presence", an online brochure of sorts, will take relatively little time and attention — it may need nothing more than an occasional sprucing up (to avoid looking like it hasn't been touched since 1998) and updating the contact information (address, telephone number, etc.) as necessary. If the information changes a lot, though, or the site is more interactive, it may take a considerable investment in time and energy both to do the work and to stay abreast of the knowledge needed to do that work. Even a "point of presence" that is based on a content management system (like LightCMS, WordPress, Joomla or Drupal) will need continual updates to patch security holes as they're discovered.
As for ease of migration, that is actually more than one question. It is only partly a technical question. It's also a legal question. You may not actually hold the necessary rights to migrate your existing website. The data are, of course, naturally yours, but the copy text (that is, the way in which those data were expressed), any logos and the website design itself naturally belong to their respective authors/creators, and your rights in those works depend on the contracts and circumstances under which they were created. When you're dealing with a combination developer/designer/host, it's not uncommon (though it is a bit shady) for your license to use the design elements to be contingent upon maintaining the rest of the service package. Without consulting the contracts (and the whois information for your domain) you don't even know whether you have any rights to your domain name.
Once the legal issues are decided, the technical issues are another question. For a simple site, you can certainly do all of the necessary work without the cooperation of your current service provider(s). All it takes is "Save As" in your browser and enough basic knowledge of HTML to recreate the (apparent) structure of the existing site. (Note that you don't need to get the underlying structure right; what matters is the structure that the browser sees.) Upload the files into the appropriate folders, and you're good to go. But for anything more complex, you'd need to have access to the raw information on the server, and that means having the administrative password(s) and direct access to the server. If this is a hostile migration, you may find that you'll have some trouble there (particularly if you're of a mind to skirt the copyright issue or are in the midst of money wrangling). Even if you can get all of the required files and/or databases, you may have to have in-depth knowledge of the CMS or scripting environment to make the changes needed to have the site run in your new hosting environment.
So yes, there is the possibility that starting from scratch will be part of the deal, and whether or not you can do that effectively and successfully yourself depends on how much you know and how much you're willing to learn. Again, for a "point of presence", the maintenance and management of the site is minimal. (And there is no good reason to use a CMS for a site that doesn't change much.) The design, though, is not minimal; both the visual design and the content are integral to the success of the site (and, by extension, of the business it represents).
There is software (both online and locally installable) that will let you throw together something that is technically a web site or web page with absolutely no knowledge of the underlying technology or the principles of design. Simply getting something online isn't an issue. Getting something online that adequately represents you and your business, that attracts rather than repels potential clients, is another matter entirely. You may find that getting professional help with the design is the way to go, but make sure that you are only contracting for the design (and perhaps the deployment), and that you're not tying your rights to use the design to anything else. If you decide to do it yourself, don't trust yourself any further than you can throw yourself — get other people to give you feedback on the design and content. And make sure you use people who would be willing to call you a moron to your face when you have it coming (the very definition of a good friend). The last thing you want is someone who will applaud the effort and love you for who you are anyway; your clients won't have the same warm fuzzies.
